Lecture Digital signal processing: Lecture 6 - Zheng-Hua Tan

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Lecture 6 - System structures for implementation presents the following content: Block diagram representation of computational structures, signal flow graph description, basic structures for IIR systems, transposed forms, basic structures for FIR systems. Digital Signal Processing, Fall 2006 Lecture 6: System structures for implementation Zheng-Hua Tan Department of Electronic Systems Aalborg University, Denmark zt@kom.aau.dk Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction MM4 z-transform MM3 DFT/FFT System System analysis System structure MM6 MM5 Filter design MM7, MM8 MM9, MM10 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 System implementation „ LTI systems with rational system function e.g „ b0 + b1 z −1 H ( z) = , − az −1 Impulse response | z |>| a | h[n] = b0 a nu[n] + b1a n −1u[n − 1] Linear constant-coefficient difference equation „ y[n] − ay[n − 1] = b0 x[n] + b1 x[n − 1] Three equivalent representations! How to implement, i.e convert to an algorithm or structure? Digital Signal Processing, VI, Zheng-Hua Tan, 2006 System implementation The input-output transformation x[n] Ỉ y[n] can be computed in different ways – each way is called an implementation ‰ An implementation is a specific description of its internal computational structure ‰ The choice of an implementation concerns with „ computational requirements „ memory requirements, „ effects of finite-precision, „ and so on Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Part I: Block diagram representation Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems „ „ „ „ „ Digital Signal Processing, VI, Zheng-Hua Tan, 2006 System implementation Impulse response „ h[n] = b0 a nu[n] + b1a n −1u[n − 1] y[n] = x[n] * h[n] is infinite-duration, impossible to implement in this way However, linear constant-coefficient difference equation provides a means for recursive computation of the output „ y[n] − ay[n − 1] = b0 x[n] + b1 x[n − 1] y[n] = ay[n − 1] + b0 x[n] + b1 x[n − 1] Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Basic elements Implementation based on the recurrence formula derived from difference equation requires „ ‰ ‰ ‰ adders y[n] = ay[n − 1] + b0 x[n] + b1 x[n − 1] multipliers memory for storing delayed sequence values Fig 6.1 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Example of block diagram representation A second-order difference equation „ y[n] = a1 y[n − 1] + a2 y[n − 2] + b0 x[n] H ( z) = b0 − a1 z −1 − a2 z − Demonstrates the complexity, the steps, the amount of resources required Digital Signal Processing, VI, Zheng-Hua Tan, 2006 General Nth-order difference equation N M y[n] = ∑ ak y[n − k ] + ∑ bk x[n − k ] k =1 k =0 M H ( z) = v[n] ∑b z k =0 N −k k − ∑ ak z − k k =1 A cascade of two systems! X[n]Ỉv[n], v[n]Ỉy[n] Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Rearrangement of block diagram „ 10 A block diagram can be rearranged in many ways without changing overal function, e.g by reversing the order of the two cascaded systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 System function decomposition v[n] M H ( z) = ∑b z k =0 N −k k − ∑ ak z − k k =1 ⎛ ⎞ ⎜ ⎟ M ⎜ ⎟⎛⎜ b z − k ⎞⎟ = H ( z ) H1 ( z ) = ∑k ⎠ N ⎜ − k ⎟⎝ k = ⎜ − ∑ ak z ⎟ ⎝ k =1 ⎠ ⎛ ⎞ ⎜ ⎟ M ⎛ − k ⎞⎜ ⎟ = H1 ( z ) H ( z ) = ⎜ ∑ bk z ⎟ N ⎜ ⎝ k =0 ⎠ − a z −k ⎟ ⎜ ∑ k ⎟ ⎝ k =1 ⎠ w[n] 11 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 In the time domain N M k =1 k =0 y[n] = ∑ ak y[n − k ] + ∑ bk x[n − k ] M ⎧ v [ n ] bk x[n − k ] = ∑ ⎪⎪ k =0 ⎨ N ⎪ y[n] = ∑ ak y[n − k ] + v[n] ⎪⎩ k =1 N ⎧ w [ n ] ak w[n − k ] + x[n] = ∑ ⎪⎪ k =1 ⎨ M ⎪ y[n] = ∑ bk w[n − k ] ⎪⎩ k =0 12 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Minimum delay implementation „ One big difference btw the two implementations concerns the number of delay elements N +M max( N , M ) 13 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Direct form I and II „ Direct form I as shown in Fig 6.3 ‰ „ Direct form II or canonic direct form as shown in Fig 6.5 ‰ 14 A direct realization of the difference equation There is a direct link between the system function (difference equation) and the block diagram Digital Signal Processing, VI, Zheng-Hua Tan, 2006 An example „ 15 Direct form I and direct form II implementation + z −1 H ( z) = − 1.5 z −1 + 0.9 z − Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Part II: Signal flow graph description „ „ „ „ „ 16 Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Signal flow graph (SFG) „ „ As an alternative to block diagrams with a few notational differences A network of directed branches connecting nodes variable 17 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Signal flow graph 18 Nodes in SFG represent both branching points and adders (depending on the number of incoming branches), while in the diagram a special symbol is used for adders and a node has only one incoming branch SGF is simpler to draw Digital Signal Processing, VI, Zheng-Hua Tan, 2006 From flow graph to system function Fig 6.12 w1[n] = w4 [n] − x[n] w2 [n] = αw1[n] w3 [n] = w2 [n] + x[n] w4 [n] = w3 [n − 1] y[n] = w2 [n] + w4 [n] Not a direct form, „ ‰ ‰ cannot obtain H(z) by inspection But can write an equation for each node „ w4 [ n ] = w3 [ n − 1] involve feedback, difficult to solve „ 19 By z-transform Æ linear equations Digital Signal Processing, VI, Zheng-Hua Tan, 2006 From flow graph to system function ⎧W1 ( z ) = W4 ( z ) − X ( z ) ⎪ ⎪W2 ( z ) = αW1 ( z ) ⎪ ⎨W3 ( z ) = W2 ( z ) + X ( z ) ⎪ −1 ⎪W4 ( z ) = z W3 ( z ) ⎪⎩Y ( z ) = W2 ( z ) + W4 ( z ) ⎧W2 ( z ) = α (W4 ( z ) − X ( z )) ⎪ ⎪ ⎨ −1 ⎪W4 ( z ) = z (W2 ( z ) + X ( z )) ⎪Y ( z ) = W ( z ) + W ( z ) ⎩ ⎛ z −1 − α ⎞ ⎟ X ( z) Y ( z ) = ⎜⎜ −1 ⎟ ⎝ − αz ⎠ z −1 − α If α is real, the system is ? H ( z) = − αz −1 Causal! h[n] = α n −1u[n − 1] − α n +1u[n] 20 All-pass Digital Signal Processing, VI, Zheng-Hua Tan, 2006 10 From flow graph to system function „ Fig 6.13 „ Fig 6.12 Different implementations, different amounts of computational resources 21 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Part III: Basic structures for IIR systems „ „ „ „ „ 22 Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 11 Direct form I N k =1 M M y[n] = ∑ ak y[n − k ] + ∑ bk x[n − k ] k =0 H ( z) = ∑b z k =0 N −k k − ∑ ak z − k k =1 23 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Direct form II N y[n] = ∑ ak y[n − k ] + ∑ bk x[n − k ] k =1 M M k =0 H ( z) = ∑b z k =0 N −k k − ∑ ak z − k k =1 24 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 12 Example H ( z) = 25 + z −1 + z −2 − 0.752 z −1 + 0.125 z − Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Cascade form Factor the numerator and denominator polynomials M H ( z) = ∑ bk z − k k =0 N − ∑ ak z − k k =1 =A M1 M2 k =1 N1 k =1 N2 k =1 k =1 ∏ (1 − f k z −1 )∏ (1 − g k z −1 )(1 − g k* z −1 ) ∏ (1 − ck z −1 )∏ (1 − d k z −1 )(1 − d k* z −1 ) b + b z −1 + b2 k z −2 H ( z ) = ∏ k 1k −1 − a2 k z − k =1 − a1k z Ns 26 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 13 An example: from 2nd-order to 1st-order cascade H ( z) = 27 + z −1 + z −2 (1 + z −1 ) ⋅ (1 + z −1 ) = − 0.752 z −1 + 0.125 z − (1 − 0.5 z −1 ) ⋅ (1 − 0.25 z −1 ) Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Parallel form by partial fraction expansion Np N1 Ak −1 k =1 − c k z H ( z) = ∑ Ck z −k + ∑ k =0 N2 Bk (1 − e k z −1 ) k =1 (1 − d k z −1 )(1 − d *k z −1 ) +∑ 28 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 14 Feedback in IIR systems Feedback loop: a closed path Necessary but not sufficient condition for IIR system (Feedback introduced poles could be cancelled by zeros) a nu[n] H ( z) = y[n] = ay[n] + x[n] y[n] = x[n] /(1 − a) 29 − a z −2 = + az −1 −1 − az All loops must contain at least one unit delay element Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Part IV: Transposed forms „ „ „ „ „ 30 Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 15 Transposed form for a first-order system Flow graph reversal or transposition also provides alternatives: reversing the directions of all branches and reversing the input and output Resulting in same H(z) H ( z) = 31 1 − az −1 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Transposed direct form II and direct form II The transposed direct form II implements the zeros first and then the poles, being important effect for finite-precision existing 32 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 16 Part V: Basic structures for FIR systems „ „ „ „ „ 33 Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Direct form „ „ „ 34 So far, system function has both poles and zeros FIR systems as a special case Causal FIR system function has only zeros (except for poles as z=0) M ⎧b , n = 0,1, , M y[n] = ∑ bk x[n − k ] h[n] = ⎨ n k =0 ⎩0, otherwise Form I and form II are the same Digital Signal Processing, VI, Zheng-Hua Tan, 2006 17 Cascade form Factoring the polynomial system function M H ( z ) = ∑ h[n]z n =0 35 −n Ms = ∏ (b0 k + b1k z −1 + b2 k z − ) k =1 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Linear-phase FIR systems Generalized linear-phase system „ H (e jω ) = A(e jω )e − jωα + jβ A(e jω ) is a real function of ω , α and β are real constants Causal FIR systems have generalized linear-phase if h[n] satisfies the symmetry condition „ h[ M − n] = h[n], n = 0,1, , M or h[ M − n] = −h[n], n = 0,1, , M 36 M y[n] = ∑ bk x[n − k ] k =0 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 18 Linear-phase FIR systems if M is an even integer M y[n] = ∑ h[k ]x[n − k ] k =0 = M / −1 ∑ h[k ]x[n − k ] + h[k / M ]x[n − M / 2] + k =0 = M ∑ h[k ]x[n − k ] k = M / +1 M / −1 M / −1 k =0 k =0 ∑ h[k ]x[n − k ] + h[k / M ]x[n − M / 2] + ∑ h[M − k ]x[n − M + k ] if h[ M − n] = h[n] y[n] = M / −1 ∑ h[k ]( x[n − k ] + x[n − M + k ]) + h[k / M ]x[n − M / 2] k =0 37 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Linear phase FIR systems M is an even integer and h[M-n]=h[n] y[n] = M / −1 ∑ h[k ]( x[n − k ] + x[n − M + k ]) + h[k / M ]x[n − M / 2] k =0 38 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 19 Discussions „ „ „ 39 Implementation of FIR and IIR systems Use signal block diagram flow graph representation to show the computational structures Although two structures may have equivalent inputoutput charateristics for infinite-precision represenations of coefficients and variables, they may have dramatically different behaviour when the numerical precision is limited Digital Signal Processing, VI, Zheng-Hua Tan, 2006 Summary „ „ „ „ „ 40 Block diagram representation of computational structures Signal flow graph description Basic structures for IIR systems Transposed forms Basic structures for FIR systems Digital Signal Processing, VI, Zheng-Hua Tan, 2006 20 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction MM4 z-transform MM3 41 DFT/FFT System System analysis System structure MM6 MM5 Filter design MM7, MM8 MM9, MM10 Digital Signal Processing, VI, Zheng-Hua Tan, 2006 21 ... diagram Digital Signal Processing, VI, Zheng-Hua Tan, 20 06 An example „ 15 Direct form I and direct form II implementation + z −1 H ( z) = − 1.5 z −1 + 0.9 z − Digital Signal Processing, VI, Zheng-Hua. .. VI, Zheng-Hua Tan, 20 06 Direct form II N y[n] = ∑ ak y[n − k ] + ∑ bk x[n − k ] k =1 M M k =0 H ( z) = ∑b z k =0 N −k k − ∑ ak z − k k =1 24 Digital Signal Processing, VI, Zheng-Hua Tan, 20 06 12... systems Digital Signal Processing, VI, Zheng-Hua Tan, 20 06 20 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction MM4 z-transform
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